What is the solution for y^{''}-5y^'+6y=2x^2-1 ?

The solution for y^{''}-5y^'+6y=2x^2-1 is y=c_{1}e^{3x}+c_{2}e^{2x}+(x^2)/3+(5x)/9+5/27

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y^{''}-5y^'+6y=2x^2-1

y^{^{''}} - 5 y^{^{'}} + 6 y = 2 x^{2} - 1

y = c_{1} e^{3 x} + c_{2} e^{2 x} + \frac{x ^{2}}{3} + \frac{5 x}{9} + \frac{5}{27}

Solution steps

y^{''} (x) - 5 y^{'} (x) + 6 y = 2 x^{2} - 1

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{2 x} + \frac{x ^{2}}{3} + \frac{5 x}{9} + \frac{5}{27}

y = c_{1} e^{3 x} + c_{2} e^{2 x} + \frac{x ^{2}}{3} + \frac{5 x}{9} + \frac{5}{27}

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} = - 3

y^{^{''}} + 4 y = 1, y (0) = 0, y^{^{'}} (0) = 1

x^{^{''}} + 4 x = 12 cos (t)

\frac{d ^{2} y}{d x ^{2}} - 3 \frac{d y}{d x} + 2 y = x^{2}

y^{^{''}} + 4 y = 4598 cos (2 x)

What is the solution for y^{''}-5y^'+6y=2x^2-1 ?
The solution for y^{''}-5y^'+6y=2x^2-1 is y=c_{1}e^{3x}+c_{2}e^{2x}+(x^2)/3+(5x)/9+5/27